Spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs

TL;DR

This paper investigates the spectral properties of adjacency and Laplace operators on percolation subgraphs of amenable Cayley graphs, focusing on the asymptotic behavior of the integrated density of states near the spectral minimum.

Contribution

It provides new results on the spectral asymptotics of percolation Hamiltonians on amenable Cayley graphs, including the behavior of the spectral distribution function.

Findings

Asymptotic behavior of the integrated density of states near the spectral minimum

Description of spectral properties of adjacency and Laplace operators on percolation subgraphs

Formulation of new results and proof strategies for spectral asymptotics

Abstract

In this paper we study spectral properties of adjacency and Laplace operators on percolation subgraphs of Cayley graphs of amenable, finitely generated groups. In particular we describe the asymptotic behaviour of the integrated density of states (spectral distribution function) of these random Hamiltonians near the spectral minimum. The first part of the note discusses various aspects of the quantum percolation model, subsequently we formulate a series of new results, and finally we outline the strategy used to prove our main theorem.